A Comparison of Different Models for Probabilistic Kick Tolerance Calculations

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FACULTY OF SCIENCE AND TECHNOLOGY

MASTER'S THESIS

Study program/specialization:

Petroleum engineering/

Drilling and Well engineering

Spring semester, 2020 Open/Confidential

Author:

Martine Kristoffersen

(signature of author)

Supervisor(s):

Kjell Kåre Fjelde Title of master's thesis:

A Comparison of Different Models for Probabilistic Kick Tolerance Calculations Credits (ECTS): 30

Keywords:

Kick tolerance Well control Well design Flow models

Monte Carlo Simulations MATLAB

Number of pages: 62

+ supplemental material/other: 36 Stavanger, June 15^th 2020

Title page for Master's Thesis Faculty of Science and

Technology

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Acknowledgements

First, I would like to express my sincere gratitude to my supervisor Kjell Kåre Fjelde for incredible support and involvement during the work of this thesis. Thank you for providing me with an exciting problem, and for great follow-up throughout my work.

Secondly, I wish to thank my amazing friends and co-students for motivation and help over the course of the last few years. I could not have done this without you!

Lastly, a huge thank you to my family, and especially Magnus, for endless support and for cheering me on throughout this degree and every other aspect of my life.

14^th of June, 2020

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III

Abstract

Kick tolerance is an evaluation of how large kick sizes an open hole section can withstand without threatening the formation integrity at the shoe. If a certain kick size cannot be handled safely, the planned open hole section needs to be shortened, and the casing design has to be altered.

Three different flow models were used to simulate a kick situation: single bubble, transient flow, and analytical. The single bubble model and the transient flow model, based on the drift flux model, describe the kick circulation in time while the analytical model is a static model where the kick is situated at the bottom. These are all based on gas kicks in water-based mud. The simulations were performed with MATLAB, and to generate the different results, length and size of the open hole, BHA length and kick size are varied.

First, a transient model based on the single bubble concept was considered, and a Monte Carlo simulation framework was implemented. This modification was done along with extensive coding in order to further develop the previously written code. The purpose of the simulations was to see what kick location would give the largest casing shoe pressure: kick at bottom hole assembly (BHA) or kick expanding whilst travelling upwards towards the shoe. The results showed that in most cases, the maximum casing shoe pressure would occur when kick is located at the BHA. However, when simulations were performed with a long hole section, short BHA and large kick volume, gas expansion caused the maximum casing shoe pressure.

Secondly, the three models were simulated, and results were compared. The output was the fracture pressure and casing shoe pressure in the form of distributions. From the many Monte Carlo simulations of each case, the result was a failure probability based on the number of counts where the casing shoe pressure exceeded the fracture pressure at the shoe. The main objective of these simulations was to see how the three models compared by looking at the failure probabilities.

When comparing the models, the results show that the single bubble model provides the most conservative results and the transient flow model the least conservative. It would be reasonable to first calculate the casing shoe pressure to identify kick location at maximum pressure. Then, if the maximum pressure is caused by gas expansion, a transient model should be used to account for this.

If not, the analytical model could be deemed appropriate. When calculating failure probability, the single bubble model might be too conservative due to the fact that it assumes the gas kick to be a single slug.

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List of Figures

Figure 2-1 A simplified pore pressure plot ... 15

Figure 3-1 Kill sheet for the driller's method [6] ... 19

Figure 3-2 Choke pressure development for the driller's method [6] ... 20

Figure 3-3 Kill sheet for the wait and weight method [6] ... 21

Figure 3-4 Choke pressure development for the wait and weight method [6] ... 21

Figure 3-5 Casing shoe pressure vs. Time ... 23

Figure 3-6 BHP development during a kick situation [6] ... 23

Figure 3-7 Kick height during kick circulation ... 24

Figure 3-8 Top and bottom position of the kick ... 25

Figure 3-9 Choke pressure development ... 26

Figure 3-10 Casing shoe pressure vs. Kick size [9, p. 10] ... 28

Figure 4-1 Wellbore sketch ... 30

Figure 4-2 Cross-sectional area around BHA ... 30

Figure 5-1 Example: Distribution of travel time ... 35

Figure 5-2 Simplification of the MCS process ... 36

Figure 5-3 Common input distribution shapes in MCS ... 38

Figure 5-4 Normal distribution with varying characteristics ... 39

Figure 5-5 Example: Triangular distribution ... 39

Figure 5-6 N=100 ... 40

Figure 5-7 N=1,000 ... 40

Figure 5-8 N=10,000 ... 41

Figure 5-9 N=100,000 ... 41

Figure 5-10 Example 1: 0.43% chance of failure ... 42

Figure 6-1 Mean values of maximum casing shoe pressure: Long hole section & 50 m BHA ... 53

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VII

Figure 6-2 Mean values of maximum casing shoe pressure: Long hole section & 150 m BHA ... 53 Figure 6-3 Mean values of maximum casing shoe pressure: Short hole section & 50 m BHA ... 54 Figure 6-4 Mean values of maximum casing shoe pressure: Short hole section & 150 m BHA ... 54

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List of Tables

Table 2-1 Common hole sizes and corresponding casings ... 14

Table 4-1 Definition of parameters ... 32

Table 6-1 Overview of parameters used in maximum casing shoe pressure simulations ... 46

Table 6-2 Long hole section & 12.25” open hole ... 47

Table 6-3 Long hole section & 8.5” open hole ... 47

Table 6-4 Short hole section & 12.25” open hole ... 47

Table 6-5 Short hole section & 8.5” open hole ... 48

Table 6-6 Fixed parameters for all three flow models ... 49

Table 6-7 Varying parameters for all three flow models ... 50

Table 6-8 Single bubble with long hole section ... 51

Table 6-9 Single bubble with short hole section ... 51

Table 6-10 Transient flow model with long hole section ... 51

Table 6-11 Transient flow model with short hole section ... 51

Table 6-12 Analytical model with long hole section ... 52

Table 6-13 Analytical model with short hole section ... 52

Table 6-14 Maximum casing shoe pressures compared ... 55

Table 6-15 Change in failure probability and CPU times for various number of MCS ... 56

Table B-1 Maximum casing shoe pressure in bar, long hole section & 12.25” hole ... 98

Table B-2 Maximum casing shoe pressure in bar, long hole section & 8.5” hole ... 98

Table B-3 Maximum casing shoe pressure in bar, short hole section & 12.25” hole ... 98

Table B-4 Maximum casing shoe pressure in bar, short hole section & 8.5” hole ... 98

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IX

List of Equations

(3.1) ... 22 (3.2) ... 27 (5.1) ... 38

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Abbreviations

BHA Bottom Hole Assembly BHP Bottom Hole Pressure BOP Blow Out Preventer CPU Central Processing Unit

DP Drill Pipe

FIT Formation Integrity Test ID Inner Diameter

LOT Leak-Off Test

MCS Monte Carlo Simulation OBM Oil-based Mud

OD Outer Diameter

PDF Probability Density Function SICP Shut-in Casing Pressure SIDPP Shut-in Drill Pipe Pressure

SM Safety Margin TVD True Vertical Depth WBM Water-based Mud WOW Wait on Weather

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Symbols

sg Specific gravity K Kelvin

bbl Barrel

μ Expected value σ Standard deviation ρgas Density of gas

ρmud Density of drilling fluid g Gravitational constant

ABHA Cross-sectional area outside bottom hole assembly ADP Cross-sectional area outside drill pipe

Pshoe Casing shoe pressure Pbh Bottom hole pressure

Pfr Fracture pressure at the shoe D Depth of well

Dshoe Depth of casing shoe Hkick Height of kick in the well

Q Flow rate

N Number of Monte Carlo simulations

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1 Introduction

1.1 Background

When constructing a well, there are several elements that need to be estimated in order to plan how the well should be built. One of these is the length of each casing section. It is optimal to make each section as long as possible to save time and cost. One of the defining factors when designing casing and hole sections is the formations ability to handle a certain kick size when we drill down to planned depth. The weakest formation point is usually just below the last set casing shoe. If formation cannot handle it, the planned hole section needs to be shortened which again impacts the length of the casing. This evaluation is called kick tolerances, which represents an evaluation of which kick sizes that can be safely circulated out of the well without threatening the formation integrity.

Often, the worst case scenario is used to design the well. Some industries, e.g. construction, have been moving more towards using probabilistic methods to calculate failure.

To save cost and time, more research into using this method should be considered. Reliability based casing design is an example of this method, where probability is used to look at casing strengths vs. survival loads [1]. The working method in this thesis can be considered a continuation of the reliability based casing design as the main principles are the same.

1.2 Statement of the problem and objective

This thesis is a continuation of the work done in OMAE2017-61391 “Probabilistic Flow Modelling Approach for Kick Tolerance Calculations” [2]. The paper compared simulation results of two flow models in a well: the transient flow model and the analytical model.

Two things that were not considered in the previously mentioned paper, which could be interesting to look into:

- In wells with very long open hole sections, will the highest casing shoe pressure always occur when the kick passes bottom hole assembly (BHA), or is it possible that gas expansion gives the highest casing shoe pressure such that the maximum pressure is achieved when the kick reaches the shoe?

- Comparing the two models with the single bubble model. How similar or different would the output be?

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All simulations performed were used to further look at how Monte Carlo simulations can be used in kick tolerance calculations:

- Is the use of Monte Carlo simulations for probabilistic calculation of kick tolerances a viable approach?

1.3 Structure of thesis

The thesis starts with presenting relevant theory, and continues with the simulation results obtained, a discussion and conclusion, and the codes used are attached in the appendix.

Chapter 2-5 makes up the theory:

- Chapter 2 gives a short introduction to well design and what lies behind the decisions as to how a well should be built.

- Chapter 3 dives into kick and kick tolerances. This is the most extensive theory chapter which is important to be able to understand the purpose and method for the simulations done.

- Chapter 4 presents the three models used to calculate kick tolerances. The single bubble model is weighted, as this is the main focus in the simulations. However, the two other models are also presented with their basic principles, advantages, and limitations.

- Chapter 5 explains the concept of Monte Carlo simulations. It contains information about what Monte Carlo simulations are, how they work, and their potential use in the petroleum industry.

Chapter 6 presents the data and results obtained. It contains results about where the maximum casing shoe pressure will be found when using the single bubble model. And it also shows failure probability and mean casing shoe pressure values for different scenarios and for all three models.

Conclusion and recommendations for future work are provided in chapter 7.

Lastly, the reference list and appendix is provided. Appendix A includes all codes that were used to create the results obtained in this thesis. The codes were originally developed in the work related to paper OMAE2017-61391 “Probabilistic Flow Modelling Approach for Kick Tolerance Calculations” [2]. Appendix B contains additional data obtained during simulations with the single bubble model.

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2 Well Design

A well consists of a hole of various sizes with corresponding casings which are cemented together with the formation. The hole and casing sizes are largest at seabed and they decrease in size as the well gets deeper. There are some common sets of casing and hole sizes which are widely used. Table 2-1 below gives an overview of these.

Casing type Hole size (diameter) Casing size (diameter)

Conductor 36 in 30 in

Surface casing 26 in 20 in

Intermediate casing 17 ½ in 13 ⅜ in

Production casing 12 ¼ in 9 ⅝ in

Production liner 8 ½ in 7 in

Table 2-1 Common hole sizes and corresponding casings

To determine where the casing shoe of each section should be set, a pore pressure and formation strength prognosis is used. An example of this prognosis is shown in Figure 2-1. This figure shows pressure in specific gravity (sg) plotted against true vertical depth (TVD). The idea is to use the drilling fluid density to create a hydrostatic pressure. The hydrostatic pressure is used to keep pressure inside the well above the pore pressure to avoid unwanted influx, whilst still being below the fracture pressure to avoid fracturing the formation.

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Figure 2-1 A simplified pore pressure plot

In the plot, a vertical straight line is drawn from seabed depth down until it reaches the pore pressure limit. To avoid going below the pore pressure, a casing shoe needs to be set in order to switch to a heavier drilling fluid. A horizontal line can be drawn right towards the fracture pressure curve to move away from pore pressure curve. Then we can repeat the first step by again drawing a vertical line downwards towards the pore pressure curve. This is continued downwards until target depth is reached. The result is a zig-zag line in the pore pressure plot which suggests casing shoe depths at the depth of each horizontal line section.

After a casing shoe for one section is set, pressure testing is done before drilling the next section of open hole. Usually, either a formation integrity test (FIT) or a leak-off test (LOT) is performed. A LOT is performed by pressurizing the shoe and formation until formation fracture occurs. The leak-off pressure is noted, and is used to calculate LOT at the shoe. A FIT is performed by increasing the bottom hole pressure (BHP) to designed pressure. Surface pressure is then increased until the required pressure is reached, and then the test is ceased. It is not continued until fracture, unlike the LOT, because it is only necessary to verify that the next

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section can be drilled and that the formation at the shoe can handle the pressures that will occur in a kick situation.

To save cost and time, it is most feasible to have sections as long as possible. How far it is possible to drill before setting a new casing shoe, is determined by the loads which can occur during the drilling process. The casing can be subjected to axial, burst and collapse loads. Burst loads can occur when casing is filled with gas, and can cause the casing to rupture. Collapse loads can occur during mud loss scenarios due to a reduction in hydrostatic pressure in the well.

Axial loads will occur due to the weight of the casing string, and will increase if the section is extended. Mechanical friction will come in addition. An overview of different load scenarios can be found in “Modern Well Design” by Bernt S. Aadnøy [3].

Special care must be devoted to kick scenarios that can occur. During circulation of a kick, the pressure in the well will increase, and this can threaten the weakest part of the formation. This can e.g. lead to underground blowouts. The maximum well pressure at the weakest part of the formation will occur when kick is at the bottom or when it passes the shoe. Hence if a certain kick size cannot be handled safely, the planned hole section needs to be shortened [4]. This analysis is often termed kick tolerance evaluations, and will be explained in more depth in the next chapter.

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3 Well Control and Kick Tolerances

3.1 Kick

Kick is the term used for unwanted fluids entering the well during well operations e.g. drilling or completion. It is likely to occur when the pore pressure in the formation surrounding the well is higher than the pressure in the well. The influx fluid could be water, oil, gas, or a mixture of these.

3.1.1 Causes of kicks

(1) Insufficient mud weight

The fluid column of drilling mud is used to create a hydrostatic pressure inside which is larger than the pore pressure, in order to prevent a kick. To determine the correct mud weight, the formation pressure needs to be predicted. However, during drilling it is possible that the mud weight in reality is lower than planned due to either pressure and temperature effects or wrongful weighting, which could cause a lower pressure inside the well [5].

(2) Uncertainty in pore pressure

The pore pressure can be underestimated when drilling an exploration well in a new area, due to the fact that the pressure is only a calculated estimate and there are no comparable pressure data from the area. Pore pressure is used to determine the density of drilling fluid to be used, and when the pore pressure is in fact higher than calculated, the mud will not create enough of a hydrostatic pressure to keep fluids from entering the well.

(3) Swabbing effect

Swab effects cause a decrease in the bottom hole pressure, which again can lead to unwanted influx [2].

(4) Lost circulation

If for some reason drilling fluids are lost to the formation, a drop in the annulus mud level will occur. This will lead to a drop in the bottom hole pressure, which can cause an influx from the outside permeable formation.

(5) Insufficient refill of well while tripping

Insufficient hole fill is one of the most common causes to kicks. It is usually a result of human error where the hole is not filled properly or failing to notice that it is not properly filled [5].

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3.1.2 How kick is detected

There are a few signs which can lead to early detection of a kick:

(1) Increase in pit level

An increase in pit volume at the surface is a good indicator of a kick being taken. This increase will happen because formation fluids entering the well will be added to the existing well fluids.

(2) Reduced drill pipe weight (3) Pump pressure changes

The pump pressure decreases due to a reduction in hydrostatic pressure in the annulus.

(4) Gas, oil, water-cut mud

(5) Unexpected increase in drilling rate

This can occur if the drill bit encounters a porous formation which can contain formation fluids.

This can be considered as the earliest sign of a kick [5].

3.1.3 WBM vs. OBM

Kick detection is different for oil-based mud (OBM) and water-based mud (WBM).

Water-based mud:

Gas kick is easily detected in WBM due to the fact that it will not be dissolved in the drilling fluid.

The gas kick is expected to reach the surface faster in WBM than it would in OBM since free gas has slippage relative to liquid. In addition, the kick will migrate upwards after the well has been shut in, and the well pressures will continue to increase. It will come to a stop when the kick is located just below the blow out preventer (BOP). The kick distribution will change during the migration [6].

Oil-based mud:

If the well is under high pressure, a gas kick will dissolve in an oil-based drilling fluid. This can lead to the kick not being detected as early as it would in a WBM. When the kick is dissolved, it will not migrate upwards. The kick will boil out rapidly in the upper parts of the well, and it needs immediate action due to the large expansion [6].

3.1.4 Process of well control after kick detection

If a kick is not controlled, the disastrous result could be a blowout. After the kick has been detected, action needs to be taken in order to safely circulate the kick out of the well.

The common procedure can be compiled into five steps [6]:

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19 (1) Stop pumps and rotation

(2) Close the BOP

Wait for the pressure inside the well to build up. This will continue as long as formation fluids are entering the wellbore. The kick will start to slowly migrate upwards.

(3) Monitor the shut-in pressures

One can note the shut-in drill pipe pressure (SIDPP) and the shut-in casing pressure (SICP) and use this information to estimate the pore pressure and get an idea about the kick size.

(4) Open choke and circulate the kick out through choke line to separator/flare There are two main methods to circulate a kick out of a well:

The first is the driller’s method and the second is the wait and weight method.

Driller’s Method:

This method first circulates the kick out of the well, and then proceeds to circulate kill mud.

Previously used mud is circulated through the well in order to remove the kick from the wellbore.

Then, kill mud is pumped down and circulated through the well. After this process, the well will be balanced and the pressure under control.

The kill sheet and choke pressure development is showed in Figure 3-1 and Figure 3-2.

Figure 3-1 Kill sheet for the driller's method [6]

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Figure 3-2 Choke pressure development for the driller's method [6]

Wait and Weight:

This method circulates the kick out whilst circulating the kill mud. While the well is shut in, density of the mud is increased to kill mud weight, and pumped down. This means that the well will be killed with one single circulation, unlike driller’s method which takes two circulations [5].

The kill sheet and choke pressure is showed in Figure 3-3 and Figure 3-4.

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Figure 3-3 Kill sheet for the wait and weight method [6]

Figure 3-4 Choke pressure development for the wait and weight method [6]

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(5) Bullheading

As a last resort, the kick can be bullheaded. This involves forcing the formation fluids back into the formation by using a heavy weighted mud. Bullheading can be an alternative in situations where the kick volume is substantial enough that it can break the formation on its way up the well, or when H2S is present [6].

3.2 Pressure development in a well during kick circulation 3.2.1 Casing shoe pressure development

When using a transient model to calculate the casing shoe pressure, the pressure development can look something like the example in Figure 3-5. To explain the pressure development, this example is used. The pressure starts at approximately 565 bar, which is the initial pressure before the kick is present. As the kick enters the well at the bottom, it starts by taking up the volume behind the BHA.

This will dramatically increase the casing shoe pressure due to the small cross-sectional area around the BHA. The pressure will continue to increase until the first peak at approximately 581 bar. As the kick continues to migrate upwards in the well, and it passes the BHA, the pressure will decrease due to a decrease in the kick height. This height decrease is caused by an increase in the volume as the drill pipe has a smaller cross-sectional are than the BHA. After a few hundred seconds, the pressure will again start to increase. This build up is a consequence of the gas expanding as it moves towards the shoe. The pressure continues to increase until it reaches the second peak, which occurs when the kick is located directly below the casing shoe. The casing shoe pressure will rapidly decrease towards the initial pressure when the entire kick has passed the shoe.

To calculate the casing shoe pressure, Eq. (3.1) below is used if the kick is below the shoe, and considered as a single slug [2]:

𝑃_"#$ = 𝑃_&'− (𝐷 − 𝐻_,-",− 𝐷_$'./) ∗ 𝑔 ∗ 𝜌₄₅₆ − 𝐻_,-", ∗ 𝑔 ∗ 𝜌_7#$ (3.1) Where Pcas is the casing shoe pressure, Pbh is the bottom hole pressure, D is the total depth of the well and Dshoe is the shoe depth, g is the gravitational constant, ρmud and ρgas are the mud and gas densities, and Hkick is the height of the kick. This equation is based on basic physic, where the casing shoe pressure is equal to the bottom hole pressure, minus the hydrostatic pressure caused by the mud and the kick below the shoe.

When designing a well, it is necessary to know what the casing shoe pressure development will look like if a kick occurs. The value of the maximum casing shoe pressure and also the location of

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Figure 3-5 Casing shoe pressure vs. Time

When circulating out a kick it is desired to keep the BHP constant, and this can be done by using the choke line. The choke is used to keep the BHP at pore pressure plus a safety margin (SM) [2].

This BHP development is shown in Figure 3-6.

Figure 3-6 BHP development during a kick situation [6]

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3.2.2 Simulation case

This section contains several plots to explain what happens in a well when a kick occurs. The example case used is a simulation performed with the single bubble kick simulator. More about the flow model used is covered in chapter 4.

Figure 3-7 shows the change in kick height. The initial height is zero, because no gas has entered the well. The height quickly increases to 100 m when the kick is situated behind the BHA, and the annular cross-sectional area is relatively small. As the kick passes the BHA, the area increases which leads to a decrease in the kick height. The kick height then increases exponentially due to the kick expanding whilst migrating upwards. The height decreases again as the kick gradually exits from the top of the well.

Figure 3-8 shows also shows the change in kick height, but here expressed by the top and bottom position of the kick.

Figure 3-7 Kick height during kick circulation

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Figure 3-8 Top and bottom position of the kick

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Figure 3-9 below describes the choke pressure development during the kick circulation. Because the bottom hole pressure needs the choke line in order to stay constant, the choke pressure increases as the gas expands while travelling upwards.

Figure 3-9 Choke pressure development

3.3 Kick tolerance & well design

Santos et al. defined kick tolerance like so: “Kick tolerance can be understood as the capability of the wellbore to withstand the state of pressure generated during well control operations (well closure and subsequent gas kick circulation process) without fracturing the weakest formation.” [7, p. 1].

In other words, kick tolerance is the volume of gas which can be safely circulated out of the well without damaging the formation at the weakest point. Formation at the last set casing shoe is usually considered as the weakest point.

When constructing a well, one of the most important decisions is to choose the casing setting depths.

There are many factors which affects this decision like: fractured zones, shallow gas, lithology and pore pressure. Kick tolerances will also be an important evaluation that will have an impact on the

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casing design. Implementing kick tolerance design can make the drilling process safer and more economical, and is therefore important to consider [7].

When using kick tolerances to see how deep the next casing shoe can be set, there are three main data inputs needed: pore pressure at the bottom of the well, the fracture pressure at the shoe, and the kick size that can be circulated out of the well safely [7].

To avoid fracturing the formation during circulation of a kick, Eq. (3.2) must be valid [2, p. 3]:

𝑃₈₉ ≥ 𝑃_$'./ + 𝑆𝑀 (3.2)

Where Pfr is the formation fracture pressure at the shoe, Pshoe is the casing pressure at the shoe, and SM is the safety margin.

There are two different methods to determine the maximum casing shoe depths. The first one takes into account the shut-in tolerance, which is the result when calculating the tolerance at well shut- in. The other one calculates the kick tolerance when circulating the well. This method predominantly uses a kick simulator when calculating the casing shoe pressure [8].

More about the models that make up the kick simulators will follow in the next chapter.

Usually, companies have their own set standard as to how large kick volumes the formation should be able to handle to continue drilling. What the value of this number is varies from company to company as there is no regulation as to the volume of kick that needs to be handled at each shoe.

Figure 3-10 below shows what the result of a kick tolerance evaluation might look like. This plot shows which kick sizes that can be handled for each of the pore pressure values (1.75, 1.77, 1.78 sg). When the casing shoe pressure exceeds the fracture pressure, the formation might fracture, and the kick size cannot be considered manageable. The pressure values in this plot shows that when the pore pressure is high (1.78 sg curve), the kick volume that can be safely circulated out of the well is smaller than that of a lower pore pressure.

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Figure 3-10 Casing shoe pressure vs. Kick size [9, p. 10]

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4 Models for Calculating Kick Tolerances

Different models can be used to simulate how the pressure changes in a well when a kick is encountered. They differ due to the assumptions made to make up each model. In this thesis, two of the models used are transient, and one is not. Only two of the models include a gas fraction which describes the relationship between mud and kick in the cross sectional area. These factors can have a great impact on the result, and therefore choosing a model is an important part of the simulation process.

In this thesis, the three models that are considered are as follows: single bubble model, transient flow model, and analytical model. They are described in more detail below. The single bubble model is described in more depth compared to the others due to the extensive work that has been done with the script, and also due to this model being the main focus as the other two models are included in order to compare them against the single bubble model.

4.1 Assumptions Well geometry

All three models are based on the same simplified well geometry. It consists of a wellbore with a constant inner diameter, a BHA at the bottom of the well, and a drill string from the top of the well down to the BHA. Figure 4-1 and Figure 4-2 show what the wellbore looks like. The depths and lengths of the components can be varied depending on which case is to be looked at. The well is considered to be completely vertical.

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Figure 4-1 Wellbore sketch Figure 4-2 Cross-sectional area around BHA

Gas

For the work in this thesis, the kick is considered to be made up of methane gas only, which would correspond to the worst case scenario in the simulations. The density is calculated by using the real gas law, and it will change depending on the location of the kick.

Mud

All three models are constructed for the use of water-based mud.

The transient models can be modified in order for it to be possible to simulate with oil-based mud.

The reason for the need of modification is that the kick will dissolve partially or completely in oil- based mud. Also the kinetics should be taken into account to include that the dissolution process takes time [10].

Temperature

The temperatures at the bottom and top of the well are set as constant. A linear temperature gradient is created from these two boundary temperatures.

4.2 Single bubble model

The most significant assumption to this model is based on the kick being a single bubble, meaning the gas fraction is equal to one and that the gas takes up the entire cross-sectional area of the wellbore. In reality, a gas fraction of one is unlikely to happen as the distribution between gas and

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31 The script calculates depending on the kick location:

1) Kick is below the casing shoe a) Kick is only around BHA

b) Kick is in transition zone between BHA and drill pipe (DP), or only around DP

The kick height dramatically changes due to a change in the kicks cross-sectional area from BHA to drill pipe.

2) Kick is passing the shoe

This is the transition zone where kick will move from being entirely below the shoe to being above the shoe. This will lead to a decrease in casing shoe pressure.

3) Kick is entirely above the shoe.

Casing shoe pressure is no longer affected by the kick.

The equations valid for each of these situations are stated below, along with definitions of each parameter used in the equations which can be found in Table 4-1:

Parameter Definition

Vkick_initial Volume of kick just after the entire kick has entered the well [m³]

Vkick Volume of kick, time dependent [m³]

Dwell Depth of well [m]

Dshoe Depth of shoe [m]

Dkick_top Depth of the kicks top position [m]

Dkick_bottom Depth of the kicks bottom position [m]

LBHA Length of BHA [m]

DBHA Depth of BHA [m] (Depth corresponds to top of BHA) ABHA Cross-sectional area around BHA [m²]

ADP Cross-sectional area around drill pipe [m²]

Lopenhole Length of well from top to shoe [m]

Lcas Length of casing section [m]

Hkick,BHA Kick height around BHA [m]

Hno_kick,BHA Mud height around BHA [m]

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Hkick,openhole Kick height around DP, below shoe [m]

Hno_kick,openhole Mud height around DP, below shoe [m]

Hkick,cas Kick height around DP, above shoe [m]

Hno_kick,cas Mud height around DP, above shoe [m]

Habovekick Total mud height above kick [m]

Hunderkick Total mud height below kick [m]

ρgas Density of gas bubble, time dependent [kg/m³] ρmud Density of mud, constant [kg/m³]

t Time [s]

tinflux Time just after the entire kick has entered the well [s]

Q Mud rate [m³/s]

Table 4-1 Definition of parameters

1. Kick is below the casing shoe a) Vkick < LBHA*ABHA

Hkick,openhole = Hkick,cas = 0

Hno_kick,openhole = Lopenhole, Hno_kick,cas = Lcas

Hkick,BHA = Vkick/ABHA

Hno_kick,BHA = LBHA- Hkick,BHA

b) Dshoe <= Dkick_top

Vkick = Pbh*Vkick_initial*Zb*Tb/Zbhp*Tbh*Pb

Hkick,cas = 0, Hno_kick,cas = Lcas

1) If: Hkick,BHA = 0, no kick around BHA Hkick,BHA = 0, Hno_kick,BHA = LBHA

Hkick,openhole = Vkick/ADP

2) If: Hkick,BHA > 0, transition zone between BHA/DP Hno_kick,BHA = Q*t-tinflux/ABHA

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33 Vkick,BHA = Hkick,BHA * ABHA

Vkick,openhole = Vkick - Vkick,BHA

Hkick,openhole = Vkick,openhole / ADP

Hno_kick,openhole = Lopenhole - Hkick,openhole

Hkick_total = Hkick,BHA + Hkick,openhole

Habovekick = Dwell - Hkick_total - Hunderkick

Pcas = Pbh - (Habovekick - Dshoe + Hunderkick)*g* ρmud - Hkick_total*g* ρgas

2. Kick is passing the shoe

Dkick_top < Dshoe & Dkick_bottom > Dshoe

Hkick,BHA = 0, Hno_kick,BHA = LBHA

Hno_kick,openhole = Lopenhole - Hkick,openhole

Hno_kick,cas = Lcas - Hkick,cas

Pcas = Pbh - Hunderkick*g*ρmud - Hkick,openhole*g* ρgas

3. Kick is entirely above casing shoe Dkick_bottom < Dshoe

Hkick,BHA = 0, Hno_kick,BHA = LBHA

Hkick,openhole = 0, Hno_kick,openhole = Lopenhole

Hkick,cas = Vkick/ADP, Hno_kick,cas = Lcas - Hkick,cas

Pcas = Pbh - ρmud *g*Lcas

4.3 Transient flow model

The transient flow model is built on the transient drift flux model, which again is based on the conservation of both mass and momentum for a one-dimensional two-phase flow.

The model takes gas slip into account. This principle is based on the fact that gas has a higher velocity than the liquid, which in this case is mud. This will lead to changes in the gas distribution as the kick moves upwards in the wellbore.

This model is based on the usage of water-based mud, where the kick will not be dissolved in the mud. Because this is a transient model, it would be possible to simulate with oil-based mud by implementing changes in the simulation script. More information about WBM vs. OBM can be found in section 3.1.3.

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Since this is a transient model, the casing shoe pressure increases when the kick travels towards the shoe due to gas expansion. This is one of the great advantages of this model combined with the fact that the gas is distributed.

The mud density varies with time, and is a function of temperature and pressure.

Because this is a transient model and variables like gas fraction, fluid densities and pressure changes with time, the CPU times will be higher than for a non-transient model. This is probably the largest disadvantage of using this model.

For further details on this model, see OMAE2017-61391 [2, p. 5].

4.4 Analytical model

The analytical model is a static model, which means its input parameters do not depend on time. It assumes that the kick is situated at the bottom. This makes the model unable to be used in cases where the highest casing shoe pressure will occur due to gas expansion. This is a great disadvantage to the model, but will not impact the result as long as there is reason to believe that the largest casing shoe pressure occurs when the kick passes the BHA. This disadvantage will become more prominent in cases of long open hole sections as this would give the gas more time to expand towards the shoe.

The gas fraction is determined with a numerical value or distribution. This means that area where the kick is located contains both gas and mud. This gas fraction will depend on how the kick is taken in the well. By adding the gas fraction as a distribution, the uncertainty of how the kick is taken will be taken into account. More about the use of distributions will follow in the next chapter.

This is a more realistic approach than using a gas fraction equal to one like the single bubble model does.

The analytical model has the shortest CPU time of the three models. This is one of the major advantages to using this model, as simulations will be much quicker to perform compared to the other two models.

For a more detailed description of this model, see OMAE2017-61391 [2, p. 4].

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5 Monte Carlo Simulation Technique

5.1 Basic statistical concept

Monte Carlo Simulation (MCS) is a statistical tool used to include uncertainty of input values, to be able to predict the outcome in terms of both value and its probability distribution. It is a useful tool to get a probability function which represents the desired output value in a satisfactory manner [11]. In simple terms, it is done by taking input variables in terms of probability density functions, calculating output, and repeating this process a set number of times (number of Monte Carlo simulations). The result will be a probability density function of the output.

One simple example of the use of Monte Carlo simulations is when predicting the travel time of a bus. There are several uncertainties to the travel time such as: amount of stops due to passengers getting off, stops due to passengers stepping on, acceleration and braking times, and also traffic at the time. Travel time can be calculated in this case by making a distribution for each input parameter, and using Monte Carlo simulations to combine these resulting in one distribution which shows the probability of each travel time occurring. All columns add up to 100% probability, which means that all likely outcomes are represented in the distribution. A suggestion as to what this simulation result might look is presented in Figure 5-1 below.

Figure 5-1 Example: Distribution of travel time

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5.2 Monte Carlo simulation process

Figure 5-2 simply describes the process of using Monte Carlo simulations. On the left hand side, the uncertainties of each input are included with a probability distribution for each of the input variables. In the Monte Carlo simulation process, where a predetermined model is applied, these uncertainties will be combined by running the simulation a set number of times, and results in a probability distribution function of the output argument. This result is shown on the right hand side of Figure 5-2.

Figure 5-2 Simplification of the MCS process

5.2.1 The five steps of MCS process

The process of Monte Carlo simulations can be divided into five main steps [12]:

5.2.1.1 Define the model

First, a mathematical model has to be defined in order to relate output with the input variables. This could be a set of equations where the output is calculated using the input variables, both constants

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E.g., when calculating the casing shoe pressure where the kick is below the shoe and a single slug, Eq. (3.1) is needed.

Secondly, one has to decide what the output of the simulation should be. This is usually clear as this is the unknown value one wants to identify. Examples of outputs can be cost or time of a well operation, which has been done in [12]. Another output example could be to compare well pressure vs. casing strength, which has been done in [13].

Thirdly, input parameters to the model has to be decided. The inputs can include both single numbers and parameters with an uncertainty distribution.

Events can also be included as input arguments. Examples of events during drilling could be kick or wait on weather (WOW). Each event has a probability distribution attached to it, and will have a certain impact on the output.

5.2.1.2 Data gathering for approximating input arguments

Choosing the right data for estimating each input argument and the uncertainty of each input is important. The data set should include several data points such that it gives a representative sample.

Also, data points which seems unlikely or out of place should be investigated, and perhaps be excluded from the data set [12].

Pore pressure is an example of an input argument which needs to be approximated when drilling a new well with no or few existing wells in close proximity. There is no way of precisely measuring the pore pressure before drilling down to the planned depth. To estimate the pore pressure before the well is drilled, data gathered from previously drilled wells in the same area or in the same field can be used.

5.2.1.3 Define input distributions

Input parameters, which have uncertainties will have an impact on the output, needs to be defined properly in order to be suitable for use in Monte Carlo simulations. The distribution shape and parameters quantifying the distribution has to be determined. Distribution shape depends on what type of input it is.

The distribution shapes determine how each input argument is distributed. Figure 5-3 shows some of the most common shapes:

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Figure 5-3 Common input distribution shapes in MCS

For cost and time estimation, the use of uniform or triangular distributions is acknowledged.

Although they might appear to be overly simplified, they can be quite accurate and the best choice in some situations [14].

In addition to the distribution shape, the parameters of the distribution have to be determined. Which distribution parameters are needed depends on which type of distribution is chosen.

For a normal distribution, these parameters are mean value (μ) and its standard deviation (σ). The mean value equals to the median value for this distribution as the shape is symmetrical.

The equation for calculating the normal distribution is given in Eq. (5.1):

𝑓(𝑥) = 1

√2𝜋𝜎𝑒^F^(GFH)

I

JK^I (5.1)

Figure 5-4 shows three examples of normal distributions with varying mean value and standard deviation.

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Figure 5-4 Normal distribution with varying characteristics

For a triangular distribution, the distribution parameters are minimum, maximum, and most likely value. Figure 5-5 shows an arbitrary triangular distribution, and how the shape is directly affected by its parameters. To specify the input distribution parameters properly and in accordance with available knowledge is crucial to be able to have any trust in the simulation results. In [12], Williamson et al. points out that choosing the input parameters properly is more important than choosing the right distribution shape.

Figure 5-5 Example: Triangular distribution

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5.2.1.4 Sample input distributions

Before using the input distributions in a Monte Carlo simulation, their quality should be evaluated.

Numbers from the input distributions must be handled with care, as they are randomly generated numbers and can have a great impact on the final result. If input variables are correlated, this will have to be dealt with in the Monte Carlo simulation process and this will affect the results [12].

5.2.1.5 Interpret and use the results

The output of the Monte Carlo simulation will be one or more probability density functions, depending on the number of outputs the simulation was created for. The result for a single output value, like travel time in the previously used example, is a probability-distribution curve. Like the input distribution, the output distribution should also be quality checked before it is used to draw any conclusions or make any decisions [12].

One aspect is to evaluate if the number of simulations chosen is sufficient. Figure 5-6 to Figure 5-9 shows a Monte Carlo simulation with the same input values. The input distribution shape of two of the input parameters is triangular. The only parameter which is changed in the four simulations is the number of Monte Carlo simulations, termed N. As demonstrated in the four figures below, the greater number of simulations, the smoother the probability curve will be. Having a sufficient number of simulations will lead to a more stable result, which means that a repetition of the simulation will generally give the same result. How many Monte Carlo simulations is necessary, depends on how accurate the result needs to be, e.g. what failure probability can be tolerated. If a 0.1% failure probability is tolerated, a variation of e.g. 0.5% for each simulation would not be sufficient.

Figure 5-6 N=100 Figure 5-7 N=1,000

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Figure 5-8 N=10,000 Figure 5-9 N=100,000

5.3 Applications of Monte Carlo for well engineering

Monte Carlo simulations can be used to predict several aspects of a well. It can be used to estimate cost and time of well construction, oil production rates, and also time management of projects [12].

An example of use of Monte Carlo simulations for probabilistic time and cost for P&A operations is shown in [15]. It can also be used for reliability based casing design where the loads which can occur in a well are compared with the casing strength in a probabilistic matter [1], [13].

Monte Carlo simulations can also be used to estimate the probability of a kick occurring [16]. It has also been used to give probabilistic estimates of potential blowout rates and volumes for environmental risk assessment [17].

One specific situation where Monte Carlo simulations can be useful is when examining if casing pressure will exceed the fracture pressure of the rock when drilling a new section. The risk of this happening increases if a kick occurs during drilling. To figure out if the casing can handle the kick, a Monte Carlo simulation can be done to see what the casing shoe pressure will be expressed as a probability density function. An appropriate model must be chosen for estimating the maximum casing shoe pressure that can occur. This is because there is more than one way, and more than one set of equations to calculate this pressure. The chosen model will depend on several uncertain input parameters. In addition, a probability distribution must be created for the fracture pressure. This is done by choosing a distribution shape and its parameters, as explained earlier in this chapter.

By adding the two functions to the same plot, where pressure is projected on the x-axis and probability on the y-axis, the overlapping area will represent the failure probability.

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Figure 5-10, Figure 5-11, and Figure 5-12 below demonstrates this concept. All three examples are simulated with the same pore pressure input, but the parameters for the fracture pressure distribution are varied. Example 1 and 3 have the same mean value but different standard deviation, and example 1 and 2 has different mean values, but the same standard deviation. The shaded areas highlight the overlapping space, previously described as the failure probability. A significant increase in failure probability can be seen both when the mean value of the fracture pressure is decreased, and when the standard deviation is increased. It is worth noting that both the casing shoe pressure and the fracture pressure must be located in the shaded area during one realization or draw in the loop representing the Monte Carlo simulation process. Hence, the shaded area can in some sense give a false visual impression of how large the probability of failure really is. During the Monte Carlo simulation process it will be counted how many times the casing shoe pressure exceeds the fracture pressure. This number is then divided by the total number of Monte Carlo simulations and multiplied by 100 to provide a percentage value of the failure probability. It is important to ensure that a sufficient number of Monte Carlo simulations is chosen so that the failure probability doesn’t vary too much when repeating the Monte Carlo simulation process.

Figure 5-10 Example 1: 0.43% chance of failure

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43

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This example of application of Monte Carlo simulations for probabilistic kick tolerance evaluations has been discussed in [2]. The work to be presented in this thesis is a continuation of that work.

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6 A Probabilistic Modelling Approach for Evaluating Kick Tolerances

The work done for this thesis can divided into two main parts, and is described in 6.1 and 6.2.

Chapter 6.1 uses the single bubble model to analyze how different parameters, like kick size and BHA length, affect the casing shoe pressure, and also where the kick is located when the highest casing shoe pressure occurs.

Chapter 6.2 implements Monte Carlo loops in the three models presented in chapter 4, and investigates how the well and kick parameters affect the casing shoe pressure in a probabilistic manner. The purpose of this part is to compare the three models, to see how similar the casing shoe pressures are, and how this will affect the probability of failure due to a kick.

Chapter 6.3 sums up the overall results related to the number of Monte Carlo simulations, and simulation times for the different models.

The simulation methods and the way of presenting the results is greatly inspired by the work done in [2].

6.1 Single bubble model – Identifying maximum casing shoe pressure

To perform simulations using the single bubble model, an existing script built on the model was provided by the supervisor of this thesis. The base script was mainly developed by Dalila Gomes, UiS, and then revised for the purpose of this thesis. Changes had to be made in order to make the script more robust, and to make it easier to change values of input parameters. Some additional lines were added to generate plots, and bugs related to generalizing of the script were fixed. The MATLAB code used is provided in Appendix A.1.1.

6.1.1 Purpose of simulation

The purpose of simulating with this single bubble script was to see what kick location would give the highest casing shoe pressure at the shoe. The analytical model does not take into account the gas expansion aspect of casing pressure. Therefore, the most interesting result would be to see if it was possible that gas expansion in some cases would cause the maximum pressure. The simulations with single bubble model were done with various conditions to see if any of them would in fact give maximum casing shoe pressure when the kick expands towards the shoe.

6.1.2 Method of simulation

The pore pressure is given, mud density remains constant, and friction is not accounted for.

Simulating with the single bubble model was done as early as in 1968 by J. L. Leblanc and R. L.

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Lewis in [18], which was one of the first times this had been done with the assumptions mentioned above.

Most parameters stayed fixed during the simulations, except these five: Shoe depth, BHA length, inner diameter (ID) of open hole and outer diameter (OD) of BHA, and lastly the kick volume. The value of each parameter is shown in Table 6-1 below.

Table 6-1 Overview of parameters used in maximum casing shoe pressure simulations

The kick is taken during circulation, as mud pump rate is taken into account. Temperature at the top and bottom is set, and a linear temperature gradient is created from these two values. Friction is not taken into account. The single bubble assumes that the cross-sectional area at kick depth is entirely taken up by gas, so that the gas fraction is equal to one. The model is based on the use of water-based mud only, which means that the kick will not be dissolve in the mud. The kick is assumed to consist of methane gas only, which will give the worst case scenario. Appendix A.2.1.

contains the code for calculating the density of the gas bubble at any time. The influx time is set to 100 seconds, which means that the kick enters the well gradually and the entire kick is in the well at time equal to 100 seconds.

Parameter Value

Fixed parameters

Bottom hole pressure (pore pressure + SM) 942 bar (932 + 10 bar)

Well depth 5000 m

OD drill string 5 in

Mud density 1920 kg/m³

Mud pump rate 350 L/min

Influx time for kick 100 s

Temperature at bottom of well 373 K

Temperature at the surface 323 K

Varied parameters

Shoe depth 3000 m/4500 m

BHA length 50 m/100 m/150 m

ID open hole & OD BHA 12.25 in & 8.5 in/8.5 in & 6.5 in

Kick volume 2 m³/4 m³/8 m³/12 m³

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47 6.1.3 Simulation results

The results are compounded into four tables. The four tables show results for long and short hole section, and 12.25 in open hole and 8.5 in open hole. Because the purpose of the simulation was to see which kick location that would give the highest casing shoe pressure, the results are presented with the location of the kick which gave maximum casing shoe pressure. They can be found in Table 6-2 to Table 6-5 below. The pressure values can be found in Appendix B.1.

Kick size

2 m³ 4 m³ 8 m³ 12 m³

BHA length

50m BHA BHA Shoe Shoe

100m BHA BHA BHA BHA

Table 6-2 Long hole section & 12.25” open hole

Kick size

2 m³ 4 m³ 8 m³ 12 m³

BHA length

50m BHA Shoe Shoe Shoe

100m BHA BHA Shoe Shoe

150m BHA BHA BHA Shoe

Table 6-3 Long hole section & 8.5” open hole

Kick size

2 m³ 4 m³ 8 m³ 12 m³

BHA length

50m BHA BHA BHA BHA

Table 6-4 Short hole section & 12.25” open hole

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Kick size

2 m³ 4 m³ 8 m³ 12 m³

BHA length

50m BHA BHA BHA *

100m BHA BHA BHA *

150m BHA BHA BHA *

Table 6-5 Short hole section & 8.5” open hole

Table 6-5 remark: For 12 m³ kick, no result has been presented. This is due to the fact that the kick size is larger than the volume of open hole in the well below the casing shoe.

As can be read from the tables, there are certain situations where gas expansion will cause the highest casing shoe pressure. Short hole sections gave maximum pressure when kick is at BHA, as could be expected. In cases of long hole section, large kick and short BHA, the results indicate that gas expansion caused the highest casing shoe pressures. This indicates that it is not sufficient to assume the highest pressure always occurs when kick is located at BHA. This is especially important to note when long hole sections and large kick sizes are considered.

6.2 Comparing kick tolerance calculation models

When comparing the three models presented in chapter 4, previously written scripts were used. For the single bubble model, the revised code used in 6.1 was further revised in order to implement a Monte Carlo simulation loop. For the two other models, scripts with Monte Carlo framework already implemented were provided by the supervisor of this thesis. These two scripts have been further revised to suit the purpose of this thesis. All codes were created by Dalila Gomes and Kjell Kåre Fjelde, among others, for the purpose of work presented in OMAE2017-61391 [2]. In addition, some supporting MATLAB scripts were provided in order for the flow modelling scripts to run. All codes used in these simulations can be found in Appendix A: A.1.2 to A.1.4 and also A.2.2.

6.2.1 Implementing Monte Carlo simulations in single bubble model

In order to compare failure probabilities of single bubble, transient flow, and analytical model, the single bubble script needed adaptation. A large portion of the script needed to be wrapped in a Monte Carlo loop to be able to calculate the casing shoe pressure and fracture pressure a set number of times. In addition, several lines needed to be added to the script e.g. new vectors for saving pressure values for each Monte Carlo loop and a new plot.

A Comparison of Different Models for Probabilistic Kick Tolerance Calculations

MASTER'S THESIS

Acknowledgements

Abstract

Table of Contents

List of Figures

List of Tables

List of Equations

Abbreviations

Symbols

1 Introduction

2 Well Design

3 Well Control and Kick Tolerances

4 Models for Calculating Kick Tolerances

5 Monte Carlo Simulation Technique

6 A Probabilistic Modelling Approach for Evaluating Kick Tolerances